Systems and methods for generating a graded lattice structure and their application to additive manufacturing

ABSTRACT

Systems and methods for generating graded lattice structures that can be used as infill for additively manufactured articles. Tailored sectioning and field-based smoothing are modified polygon, e.g., circle, packing algorithms that adjust the size of the circles based on physical field data to adapt the infill generation process to a field expected to be experienced by the article. Molecular dynamically generated lattice infill is based on force balancing a node distribution instead of a circle packing. Field data can be utilized to adjust the spacing of the node distribution according to a force balance equilibrium model that accounts for the field expected to be experienced by the article being additively manufactured. The resultant non-uniform honeycomb structures from tailored sectioning, field-based smoothing, and force-balancing robustly and efficiently address the connection issues with traditional non-uniform lattice structures.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

This invention was made with government support under Contract No.DE-AC05-00OR22725 awarded by the U.S. Department of Energy. Thegovernment has certain rights in the invention.

BACKGROUND OF THE INVENTION

Additive manufacturing (also known as 3D printing) allows complexgeometry to be manufactured, which has been a challenging task withconventional manufacturing technologies such as machining, compressionmolding, or casting. In additive manufacturing, the complexity of ageometry does not require additional processes, unlike othermanufacturing technologies. For example, compression molding orinjection molding requires a mold or a die which has been conventionallymanufactured by machining, and a geometric complexity causes high costin machining a mold. In additive manufacturing, such complexity may notinvolve additional cost, and moreover, may be able to reduce articleweight. Specifically, a complex geometry of an internal latticestructure can provide a solution to lightweight applications with a goodweight-to-volume ratio while satisfying desirable performance metrics.Various types of lattice structures exist in additive manufacturing.Lattice structures can be periodic or stochastic in nature. Some latticedesigns include the Schoen gyroid, a Schwarz diamond, an octet lattice,or the Kagome lattice, to name a few. A lattice can be formed bycreating a unit cell and replicating it periodically over a region ofinterest. Lattices can be fully three dimensional or arise in a “2+1”regime, in which a two-dimensional lattice is repeated periodicallylayer upon layer.

In additive manufacturing, 3D printed structures can be converted fromComputer Aided Design (“CAD”) to layers of a print. In general, a CADfile or other representation of the part can be obtained and exported toa stereolithographic (“STL”) file, which triangulates the articlesurface into a tessellated approximation. Slicer software can then slicethe representation of the article in the STL file into consecutivelayers, defining the print boundary for each layer. The slicer softwarecan output G-Code or other additive manufacturing instructions thatdictate the tool path or scan pattern for a 3D printer to execute andgenerate a physical manifestation of the target article. Various knownCAD packages have lattice generation methods. For example, WithinMedical (Autodesk, Inc., USA), Materialise Magics (Materialise NV),nTopology Element (nTopology, Inc., USA) and Simpleware CAD (Simpleware,Exeter, UK), as well as software plugins such as IntraLattice includethe ability to generate a lattice structure.

Honeycomb and other uniform lattice structures are widely used inadditive manufacturing. Infill generation algorithms for uniform latticestructures are provided in many different commercially available slicersoftware programs. Lattice generation is often three-dimensional butknown lattice generation methodologies in slicer software programs donot properly account for the layered nature of additive manufacturing.For example, there are several slicer software programs available on themarket for desktop 3D printers (e.g., Slic3r, Cura, and Simplify3D, toname a few) that offer two-dimensional lattice structure generation.However, they generally produce lattice structures with uniform unitsize and do not account for a graded stress field that a printed partmight incur under use. In contrast, some CAD packages include latticegeneration features that purport to account for a stress field. However,these packages generally just utilize a simple functionally gradedstructure (“FGS”) approach in which the lattice generation algorithmchanges a local wall thickness (or a bead thickness) based on a gradedfield.

Functionally graded material (“FGM”) or FGSs can be found in naturalmaterials including wood, sponges, and coral. The application of FGMs orFGSs in additive manufacturing has generally focused on changing thematerial itself gradually from material A to material B or changing thelocal wall thickness as the deposition proceeds. For example, onelattice generation method utilizes a material transition from glassfiber-filled ABS to carbon fiber-filled ABS in big area additivemanufacturing (BAAM). On the FGS front, a truss structure having anisogrid internal pattern with local thickness varying based on theloading and the boundary conditions has been provided. In anotherinstance, a square grid structure was created with varying localthickness in a beam in order to increase performance. In yet anothercase, an algorithm varies the local thickness of cellular structures toincrease stiffness. Sometimes these FGS approaches are referred to asdensity-based topology optimization.

Topology optimization is a method to minimize or reduce the volume in apart design under physical constraints, typically imposed stress andboundary conditions. Various approaches for topology optimization havebeen created, including: the hard-kill and soft-kill approaches, thebubble method, genetic algorithm methods, homogenization theory methods,and simulated annealing methods. These approaches can be useful indesigning minimal weight, manufacturable lattices, that often haveunexpected geometric and topological properties in their designs.However, they have limited applicability in additive manufacturing.

Put another way, current topology optimization in the additivemanufacturing context is basically limited to thickening local edges (orlines) with a uniform base structure. This approach is not efficient inextrusion-based additive manufacturing where infill edges are extrusionlines, and thick edges are printed by extruding multiple times at thesame location. This causes at least two significant issues: (1) thelocal thickness cannot be gradually changed (e.g., from 1, 1.2, 1.5, to2), because a given edge is printed by an integer number of extrusions(e.g., printed once or twice, but not one and a half times); and (2)non-uniform thickness of edges require jump movements of a nozzle thatcause stop and start of the extrusion process. The present disclosureaddresses these and other issues.

SUMMARY OF THE INVENTION

The present disclosure provides systems and methods for generatingnon-un-uniform graded lattice infill for additive manufacturingarticles. The non-uniform lattice cells can be scaled based on aphysical field having a non-uniform intensity distribution, such as atemperature or stress field. Additive manufacturing instructions can begenerated that, when executed on an additive manufacturing machine,generate a non-uniform, graded infill structure based on a selectedfield.

In general, one innovative aspect of the subject matter described inthis specification can be embodied in a method for fabricating anarticle, the article configured to experience, during operation of thearticle, a physical field having a non-uniform intensity over the extentof the article. The method can include generating, by a computer system,representations of layers of the article. Each layer can include aninfill portion corresponding to a field-tailored lattice having cellswith sides of the same thickness. Generating the field-tailored latticefor a layer can include: (i) circle-packing the infill portion of thelayer, such that adjacent circles are tangentially in contact, and sizesof the circles correlate to values of the intensity of the physicalfield at the circles' locations, (ii) computing an intermediate latticehaving triangular cells, such that vertices of a triangular cellcorrespond to centers of three adjacent circles of the circle-packedinfill portion, and (iii) computing the field-tailored lattice havingpolygonal cells with 4 to 8 walls, such that sides of a polygonal cellcorrespond to segments between centers of adjacent triangles of theintermediate lattice. Each layer of the article can be printed by anadditive manufacturing printer in communication with the computer systemto fabricate a respective structure embodying the correspondingfield-tailored lattice in that layer's infill portion.

In general, another innovative aspect of the subject matter described inthis specification can be embodied in a slicer computer system foradditive manufacture of an article. The slicer computer system includesmemory configured to store (i) surface data representative of a surfaceof the article, (ii) field data representative of intensity values of anon-uniform physical field corresponding to the article; (iii) a slicersoftware program; and (iv) additive manufacturing instructions for thearticle. The system also includes a processor in communication with thememory. The processor is configured to execute the slicer softwareprogram stored in memory to convert surface data and field data of thearticle into additive manufacturing instructions for fabricating anon-uniform infill lattice structure for the article. Execution of theslicer software program to generate the additive manufacturinginstructions includes (i) simulation of packing a planar regionrepresentative of an infill layer portion of the article with packingshapes (e.g., packing circles), wherein sizes of the packing shapesacross the planar region are selected based on intensity values of thenon-uniform physical field at corresponding locations of the packingshapes in the planar region representative of the infill layer portionof the article, (ii) simulation of generation of an intermediate latticestructure having a first set of polygonal cells, wherein vertices of thefirst set of polygonal cells correspond to centers of adjacent packingshapes and wherein sides of the first set of the polygonal cellscorrespond to segments between the vertices of the first set ofpolygonal cells, (iii) simulation of generation of the infill latticestructure having a second set of polygonal cells, wherein vertices ofthe second set of polygonal cells correspond to centers of adjacentpolygonal cells of the first set of polygonal cells of the intermediatelattice and wherein sides of the second set of the polygonal cellscorrespond to segments between the vertices of the second set ofpolygonal cells, and (iv) conversion of the second set of polygonalcells of the simulated infill lattice structure to additivemanufacturing instructions for printing, by an additive manufacturingprinter, a respective physical infill lattice structure embodying thecorresponding simulated infill lattice structure as the infill layerportion of the article, and (v) storing the additive manufacturinginstructions in memory.

The foregoing aspects and other embodiments can each optionally includeone or more of the following features, alone or in combination. Inparticular, one embodiment includes all the following features incombination.

In some embodiments, the method can include the field-tailored latticebeing a graded honeycomb lattice with hexagonal cells of different sidelengths. Smaller side lengths can correspond to high values of thephysical field and larger side lengths can correspond to low values ofthe physical field.

In some embodiments, the field-tailored lattice includes two or morepatches, each patch having hexagonal cells with different side lengthsamong different patches, such that smaller side lengths correspond tohigh values of the physical field and larger side lengths correspond tolow values of the physical field, and one or more transition zonesdisposed between the patches, each transition zone having polygonalcells with 4, 5, 7, or 8 sides. Each patch can have hexagonal cells ofthe same side length within the same patch.

The first set of polygon cells of the infill portion of the layer canalso be constructed to placing polygon vertices on the perimeter of theinfill portion and further by pinning some or all of those polygonvertices at designated points on the perimeter. The effect on the secondset of polygonal cells is to have edges roughly perpendicular to theinfill layer perimeter. In essence, pinning at a perimeter involvesplacing a first set of polygons at the perimeter, and placing a second,inner, set of polygons, relative to the first set. Shape-packing (e.g.,circle-packing) of the infill portion of the layer can also includeplacing boundary shapes (e.g., boundary circles) on a perimeter of theinfill portion. The perimeter can have corners, and the shape-packing(e.g., circle-packing) can include pinning particular boundary shapes(e.g., circles) to the corners of the perimeter.

In some embodiments, the shape-packing (e.g., circle-packing) caninclude incorporating the physical field using at least one of tailoredsectioning, field-based smoothing, and a combination of tailoredsectioning and field-based smoothing.

In some embodiments, the method can include the additive manufacturingprinter being in communication with the computer system. The printer canbe at least one of a big area additive manufacturing (BAAM) printer anda small-scale additive manufacturing printer. The printing can includeusing a nozzle diameter in a diameter in a nozzle-diameter range of0.05″ to 0.0.2″. The printing can include extruding material includingat least one of a plastic, a fiber composite, a ceramic, and a metal.The article being printed can be one of a wing, a propeller blade, aturbine blade, a beam, or a toe of an excavator cup.

In some embodiments, the conversion to additive manufacturinginstructions for printing the respective structure embodying thecorresponding simulated infill lattice structure in the infill portionof the article includes generation of G-Code for movement of an additivemanufacturing printer head along a toolpath and extrusion of a materialincluding at least one of a plastic, a fiber composite, a ceramic, and ametal, from the printer head along the toolpath.

In some embodiments, execution of the slicer software program togenerate the additive manufacturing instructions includes simulation ofshape packing additional planar regions representative of differentinfill layer portions of the article, simulation of generation ofadditional intermediate lattice structures based on the shape packing ofthe additional planar regions, simulation of generation of additionalinfill lattice structures based on the intermediate lattice structures,and translation of the additional simulated infill lattice structures toadditional additive manufacturing instructions for printing, by anadditive manufacturing printer, additional respective physical infilllattice structures embodying the corresponding additional simulatedinfill lattice structures as additional infill portions of the article.

In general, another innovative aspect of the subject matter described inthis specification can be embodied in a computer method of generatingadditive manufacturing instructions for an object using tailoredsectioning. The computer method can include accessing, with a computersystem from memory, object geometry data representative of geometry ofthe object. The method can include slicing, with a slicer softwareprogram executed by a computer system, the object geometry data intolayer data representative of layers of the object, at least one of thelayers including an object layer boundary and an object layer infillregion. Accessing, with a computer system from memory, field intensityvalues corresponding to a non-uniform scalar field distribution over theobject layer infill region. Accessing, with a computer system frommemory, a tailored sectioning parameter indicative of a mapping betweenthe field intensity values and two or more circle sizes. Circle packing,with a circle packing program executed by a computer system, the objectlayer infill region with packing circles to define a circle-packedinfill region, wherein a first subset of packing circles are sized asone of the two or more circle sizes and a second subset of packingcircles are sized as a different one of the two or more circle sizes bythe circle packing program according to (i) the field intensity valuesof the non-uniform scalar field distribution over the object layerinfill region at positions of the packing circles; and (ii) the tailoredsectioning parameter indicative of the mapping between the fieldintensity values and the two or more circle sizes. The method can alsoinclude triangulating, with a triangulation program executed by acomputer system, the packing circles in the object layer infill regionto generate an intermediate graph of triangular cells such that verticesof a triangular cell correspond to centers of three adjacent packingcircles of the circle-packed infill region, and dual graphing, with adual graph program executed by a computer system, the intermediate graphof triangular cells in the object layer infill region to generate anon-uniform section tailored lattice grid in the object layer infillregion including two or more uniform sections of hexagonal latticepatches of different sizes corresponding to the tailored sectioningparameter, and one or more transition zones between the two or moreuniform sections of different sized hexagonal lattice patches includingirregular polygons that provide a continuous interface between at leasttwo of the two or more uniform sections of hexagonal lattice patches ofdifferent sizes. The method can also include converting the non-uniformsection tailored lattice grid to additive manufacturing instructions forprinting, by an additive manufacturing printer, a respective physicalnon-uniform section tailored infill lattice structure embodying thenon-uniform section tailored lattice grid for the object layer infillregion, and storing the additive manufacturing instructions in memory.

The foregoing and other embodiments can each optionally include one ormore of the following features, alone or in combination. In particular,one embodiment includes all the following features in combination.

In some embodiments, the physical non-uniform section tailored latticeis a modified honeycomb lattice having patches of uniform hexagonalcells of different side lengths connected by the transition zone ofirregular polygonal cells, such that the patches of uniform hexagonalcells with different side lengths have side lengths that correspond todifferent threshold intensity values of the non-uniform scalar fielddistribution over the object layer infill region. The one or moretransition zones disposed between the patches can have polygonal cellswith 4, 5, 7, or 8 sides.

In some embodiments, circle-packing the object layer infill regionfurther includes placing boundary circles on a perimeter of the objectlayer infill region. The perimeter can have corners, and thecircle-packing of the object layer infill region further includespinning particular boundary circles to the corners of the perimeter.

In some embodiments, circle-packing the object layer infill regionincludes incorporating the physical field using field-based smoothing inaddition to tailored sectioning.

In some embodiments, printing the structure embodying the non-uniformsection tailored infill lattice structure includes extruding materialincluding at least one of a plastic, a fiber composite, a ceramic, and ametal.

In general, another innovative aspect of the subject matter described inthis specification can be embodied in a computer method of generatingadditive manufacturing instructions additively manufacturing an objectusing a field-based smoothing heuristic. The method can includeaccessing, with a computer system from memory, object geometry datarepresentative of geometry of the object, slicing, with a slicersoftware program executed by a computer system, the object geometry datainto layer data representative of layers of the object, at least one ofthe layers including an object layer boundary and an object layer infillregion. The computer method can also include accessing, with a computersystem from memory, field intensity values corresponding to anon-uniform scalar field distribution over the object layer infillregion, accessing, with a computer system from memory, a field-basedsmoothing parameter indicative of a mapping between the field intensityvalues and two or more circle sizes, circle packing, with a circlepacking program executed by a computer system, the object layer infillregion with packing circles to define a circle-packed infill region,wherein the packing circles are positioned and sized with respect to theobject layer infill region by the circle packing program according to aplurality of field-based smoothing heuristic constraints including (i)neighboring packing circles are substantially tangent, and at least oneof (ii) a subset of packing circles are boundary circles that lie alonga perimeter of the object layer infill region; and (iii) size of thepacking circles at positions in the object layer infill regioncorrespond to field intensity values from the non-uniform scalar fielddistribution at the same positions in the object layer infill region,wherein the circle packing program iteratively adjusts positions andsizes of the packing circles to search for an equilibrium that causesthe circle-packed infill region to at least partially satisfy thefield-based smoothing heuristic constraints. The computer method canalso include triangulating, with a triangulation program executed by acomputer system, the circle-packed infill region to generate anintermediate graph of triangular cells such that vertices of atriangular cell correspond to centers of three adjacent packing circlesof the circle-packed infill region, dual graphing, with a dual graphprogram executed by a computer system, the intermediate graph oftriangular cells in the object layer infill region to generate anon-uniform field-smoothed lattice grid of hexagonal cells, such thatsides of a hexagonal cell correspond to segments between centers ofadjacent triangles of the intermediate lattice, converting thenon-uniform field-smoothed lattice grid to additive manufacturinginstructions for printing, by an additive manufacturing printer, arespective physical non-uniform field-smoothed infill lattice structureembodying the non-uniform field-smoothed lattice grid for the objectlayer infill region, and storing the additive manufacturing instructionsin memory.

The foregoing and other embodiments can each optionally include one ormore of the following features, alone or in combination. In particular,one embodiment includes all the following features in combination.

The computer method can include where the physical non-uniformfield-smoothed lattice structure is a modified honeycomb latticeincluding at least a subset of neighboring hexagonal cells that smoothlytransition from one side length to a different side length. Thefield-based parameter can be a step size indicative of an amount ofchange at each iterative adjustment to circle size, circle position, ora combination thereof.

In some embodiments, circle-packing the object layer infill regionfurther includes pinning particular boundary circles to corners of theobject layer boundary.

In some embodiments, circle-packing the object layer infill regionincludes incorporating the physical field using tailored sectioning inaddition to field-based smoothing.

Printing the structure embodying the non-uniform field-smoothed infilllattice structure can include extruding material including at least oneof a plastic, a fiber composite, a ceramic, and a metal.

In general, another innovative aspect of the subject matter described inthis specification can be embodied in a method for fabricating anarticle, the article configured to experience, during operation of thearticle, a physical field having a non-uniform intensity over the extentof the article. The method can include generating, by a computer system,representations of layers of the article, each layer comprising aninfill portion, wherein a representation of each layer's infill portioncomprises a corresponding a molecular dynamically generated latticehaving cells. Generating the molecular dynamically generated lattice fora corresponding layer's infill portion can include the following steps:(i) obtaining an initial node distribution over the extent of the infillportion of the layer, (ii) force balancing the spacing between the nodesin the initial node distribution toward a force balance equilibrium togenerate a force balanced node distribution, wherein the force balanceequilibrium is adjusted based on the physical field having thenon-uniform intensity, (iii) computing an intermediate lattice havingtriangular cells, such that vertices of a triangular cell correspond tothree adjacent nodes of the force balanced node distribution, and (iv)computing the molecular dynamically generated lattice having polygonalcells with 4 to 8 walls, such that sides of a polygonal cell correspondto segments between centers of adjacent triangles of the intermediatelattice. In addition, for each layer of the article the method caninclude printing, by an additive manufacturing printer in communicationwith the computer system, a respective structure embodying thecorresponding molecular dynamically generated lattice in the layer'sinfill portion.

In some embodiments, the molecular dynamically generated lattice is agraded honeycomb lattice having hexagonal cells of different sidelengths, such that smaller side lengths correspond to high values of thephysical field and larger side lengths correspond to low values of thephysical field. Obtaining the initial node distribution over the extentof the infill portion of the layer can include selecting the number ofnodes and generating an initial node distribution based on the selectednumber of nodes. Generating the initial node distribution based on theselected number of nodes can include spreading the nodes over the extentof the infill portion of the layer such that spacing between nodes isone of at least random and regular.

In some embodiments, obtaining the initial node distribution over theextent of the infill portion of the layer includes distributing at leasta portion of nodes onto a boundary of the infill portion of the layer.The method can include pinning at least a portion of nodes onto theboundary of the infill portion of the layer such that force balancingspacing includes changing spacing between nodes by adjusting unpinnednode positions without adjusting pinned node positions.

In some embodiments, the additive manufacturing printer in communicationwith the computer system is at least one of a big area additivemanufacturing (BAAM) printer and a small-scale additive manufacturingprinter. The physical field can be at least one of a stress field and atemperature field. Printing the structure embodying the lattice caninclude extruding material including at least one of a plastic, a fibercomposite, a ceramic, and a metal.

In some embodiments, a force-balanced spacing node distributioncorresponds to a Lennard Jones potential modified by a stress factor ofthe physical field.

In general, another innovative aspect of the subject matter described inthis specification can be embodied in a slicer computer system foradditive manufacture of an article. The system can include memoryconfigured to store (i) surface data representative of a surface of thearticle, (ii) field data representative of intensity values of anon-uniform physical field corresponding to the article; (iii) a slicersoftware program; and (iv) additive manufacturing instructions for thearticle. The system can include a processor in communication with thememory. The processor can be configured to execute the slicer softwareprogram stored in memory to convert surface data and field data of thearticle into additive manufacturing instructions for fabricating anon-uniform infill lattice structure for the article. Execution of theslicer software program to generate the additive manufacturinginstructions includes (i) simulation of positioning a plurality of nodesover a planar region representative of an infill layer portion of thearticle, (ii) simulation of adjusting the spacing of the plurality ofnodes within the infill layer portion of the article wherein adjustmentsare based on intensity values of the non-uniform physical field atcorresponding locations in the planar region representative of theinfill layer portion of the article, (iii) simulation of generation ofan intermediate lattice structure having a set of intermediate latticepolygonal cells, wherein vertices of the intermediate lattice polygonalcells correspond to neighboring nodes of the plurality of nodes andwherein sides of the set of the intermediate lattice polygonal cellscorrespond to segments between the vertices of the set of intermediatelattice polygonal cells, (iv) simulation of generation of the infilllattice structure having a set of infill lattice polygonal cells,wherein vertices of the infill lattice set of polygonal cells correspondto centers of adjacent intermediate lattice polygonal cells of theintermediate lattice and wherein sides of the infill lattice polygoncells correspond to segments between the vertices of the infill latticepolygon cells, (v) conversion of the infill lattice polygonal cells ofthe simulated infill lattice structure to additive manufacturinginstructions for printing, by an additive manufacturing printer, arespective physical infill lattice structure embodying the correspondingsimulated infill lattice structure as the infill layer portion of thearticle, and (vi) storing the additive manufacturing instructions inmemory.

The foregoing and other embodiments can each optionally include one ormore of the following features, alone or in combination. In particular,one embodiment includes all the following features in combination.

In some embodiments, the simulated infill lattice structure is a gradedhoneycomb infill structure having hexagonal cells of different sidelengths, such that side lengths corresponds to intensity values of thenon-uniform physical field.

In some embodiments, the spacing of the plurality of nodes correspondsto a Lennard Jones potential modified by a stress factor of the physicalfield. The modified Lennard Jones potential can satisfy the followingequation, V=4e [(σ/r){circumflex over ( )}p−α(σ/r){circumflex over( )}q], where r is the distance between adjacent nodes, and α is astress factor corresponding to the non-uniform intensity distributionover the extent of the article of the physical field. The p can be 8 andthe q can be 6.

In some embodiments, simulation of adjusting the spacing of theplurality of nodes further includes placing boundary shapes (e.g.,circles) on a perimeter of the infill portion. The perimeter canincludes corners, and simulation of adjusting the spacing of theplurality of nodes can include pinning user-selected boundary shapes onthe perimeter and to the corners of the perimeter.

In some embodiments, the intensity values of the non-uniform physicalfield are representative of at least one of a stress field and atemperature field expected to be experienced over the extent of thearticle.

In some embodiments, the conversion to additive manufacturinginstructions for printing the respective structure embodying thecorresponding simulated infill lattice structure in the infill portionof the article includes generation of G-Code for movement of an additivemanufacturing printer head along a toolpath and extrusion of a materialincluding at least one of a plastic, a fiber composite, a ceramic, and ametal, from the printer head along the toolpath

Accordingly, the present disclosure provides a simple and effective wayto generate non-uniform infill based on a functional condition, such assimulated, experimental, or expected internal stress under loading ofthe article being additively manufactured.

These and other objects, advantages, and features of the invention willbe more fully understood and appreciated by reference to the descriptionof the current embodiment and the drawings.

Before the embodiments of the invention are explained in detail, it isto be understood that the invention is not limited to the details ofoperation or to the details of construction and the arrangement of thecomponents set forth in the following description or illustrated in thedrawings. The invention may be implemented in various other embodimentsand of being practiced or being carried out in alternative ways notexpressly disclosed herein. Also, it is to be understood that thephraseology and terminology used herein are for the purpose ofdescription and should not be regarded as limiting. The use of“including” and “comprising” and variations thereof is meant toencompass the items listed thereafter and equivalents thereof as well asadditional items and equivalents thereof. Further, enumeration may beused in the description of various embodiments. Unless otherwiseexpressly stated, the use of enumeration should not be construed aslimiting the invention to any specific order or number of components.Nor should the use of enumeration be construed as excluding from thescope of the invention any additional steps or components that might becombined with or into the enumerated steps or components. Any referenceto claim elements as “at least one of X, Y and Z” is meant to includeany one of X, Y or Z individually, and any combination of X, Y and Z,for example, X, Y, Z; X, Y; X, Z; and Y, Z.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a non-uniform, graded infill structure generated based oninternal stress under loading in accordance with an embodiment of thepresent disclosure.

FIGS. 2A-C illustrate exemplary non-uniform, graded infill structuresgenerated with different embodiments of the present disclosure.

FIG. 3 illustrates a connectivity issue of non-uniform infill generationaddressed by embodiments of the present disclosure.

FIG. 4 illustrates a uniform circle packing yielding a regulartriangulation converted to a uniform hexagonal lattice grid.

FIG. 5 illustrates a non-uniform circle packing yielding irregulartriangulation converted to a non-uniform polygonal lattice grid.

FIG. 6 illustrates an exemplary section tailored infill lattice having aboundary, a coarse area, and a fine area, and a close-up portionillustrating an exemplary transition region.

FIG. 7 illustrates exemplary field-smoothed lattices after differentnumbers of iterations of field-based smoothing.

FIGS. 8A-B show an example of boundary circles being placed along aninfill boundary and certain vertices being pinned to the corners of thatboundary.

FIGS. 9A-D illustrate exemplary field data and integration of the fielddata into different field-based smoothing embodiments.

FIGS. 10A-C show an example of incorporating both the stress field andthe polygon boundary condition constraints for several field-basedsmoothing embodiments.

FIG. 11 illustrates an exemplary un-deformed graph and correspondingdual graph.

FIG. 12 illustrates an exemplary deformed graph and corresponding dualgraph.

FIGS. 13A-B illustrate an exemplary perspective view of a large-scaleadditively manufactured domain area in accordance with an embodiment ofthe present disclosure as well as a stress field associated with thesame domain area.

FIG. 14 illustrates contrasting sizes between hexagon cells of a uniformlattice fabricated and hexagon cells of a non-uniform lattice fabricatedin accordance with an embodiment of the present disclosure.

FIG. 15 illustrates a representative block diagram of one embodiment ofan additive manufacturing system for use in connection with aspects andembodiments of the present disclosure.

FIG. 16 illustrates an exemplary force balance graph.

FIG. 17 illustrates screenshots of representative visualizations of aninitial node distribution and an output node distribution after forcebalancing is applied to the initial node distribution.

FIG. 18 illustrates screenshots of representative visualizations of aninitial node distribution with a larger number of nodes than FIG. 17 andan output node distribution after force balancing is applied to theinitial node distribution.

FIG. 19 illustrates screenshots of representative visualizations ofanother initial node distribution and an output node distribution afterforce balancing is applied to the initial node distribution.

FIGS. 20A-D illustrate integration of a stress field into a nodedistribution force balancing to generate an output node distribution,triangulation of that node distribution, and dual graphing of thetriangular graph to generate a generally hexagonal lattice according toan embodiment molecular dynamics based infill generation.

FIGS. 21A-C illustrates another example of molecular dynamics basedinfill generation beginning with a field integrated node distribution.

FIGS. 22A-B illustrate a comparison between a force balanced nodedistribution without a stress field input against a force balanced nodedistribution with an exemplary stress field, with an overlayrepresentative of the infill perimeter boundary of the article to befabricated.

FIGS. 23A-B illustrate triangulation and dual graphing of the nodedistribution of FIG. 22B along with an overlay representative of theinfill perimeter boundary of the article to be fabricated.

FIG. 24 illustrates a front representative view of an infill structuregenerated by force balancing a field influenced node distributionutilizing an embodiment of molecular dynamics based infill generationwith a first number of nodes in the node distribution.

FIGS. 25A-B illustrates front and perspective representative views of aninfill structure generated by force balancing a field influenced nodedistribution with fewer nodes than FIG. 24 utilizing an embodiment ofmolecular dynamics based infill generation.

DESCRIPTION OF THE CURRENT EMBODIMENTS

The present disclosure is generally directed to systems and methods forgenerating non-uniform lattice structures. The non-uniform latticestructures can be utilized in systems and methods for fabricating anarticle using additive manufacturing, small-scale and large-scale.

One aspect of the present disclosure is generally directed to systemsand methods of fabrication of a non-uniform lattice infill with variableunit size based on a physical field (e.g., a stress field or a thermalfield) that corresponds to the article being fabricated. That is, theunit size of cells of the lattice over the extent of the article varieswith the intensity of the physical field. For example, depending on thetype of physical field and the application, higher intensity fieldvalues generally correspond to smaller unit size lattice cells and lowerintensity field values generally correspond to larger unit size latticecells, or vice versa. One example of such a non-uniform, graded infillstructure fabricated in accordance with a method of the presentdisclosure is illustrated in FIG. 1 . Additive manufacturinginstructions to fabricate the non-uniform, graded infill structure canbe generated based on a functional condition embodied in a field (e.g.,a field indicative internal stress under loading).

One aspect of the present disclosure is generally directed to systemsand methods for combining multiple uniform lattices that have two ormore different lattice unit cell sizes to generate a linked non-uniformlattice structure, i.e., a lattice structure with multiple differentlattice unit cell sizes having a suitable linkage in-between. Thelinkage connects the edges of multiple different lattice patches havinguniform unit cell sizes that do not naturally align by generating atransition lattice patch that systematically and robustly transitionsbetween the different unit cell sizes. For example, systems and methodsof the present disclosure can generate a non-uniform lattice structurethat includes a transition patch that transitions from a first uniformlattice patch having a first unit cell size to a second lattice patchhaving a second unit cell size different from the first unit cell size.FIG. 1 shows an exemplary representation. Specifically, it depicts anadditive manufacturing layer toolpath 122 generated based on an internalstress field 120 that can be utilized to fabricate a layer of the infillstructure of an article, in this case an airplane wing. The particularstress field depicted is exemplary and can vary from application toapplication. In FIG. 1 , higher stress values (e.g., in Mises) aregenerally toward the bottom portion of the article while lower stressvalues are toward the top of the article with intermediate stress valuesover a gradient between.

FIGS. 2A-C illustrate top plan views of exemplary infill structures foradditively manufactured articles. Some embodiments of the presentdisclosure are directed to systems and methods for tailored sectioningto fabricate a section tailored infill structure for an article (see,for example FIG. 2A), some are directed to systems and methods forfield-based smoothing to fabricate a field-smoothed infill structure foran article (see, for example FIG. 2B), and some embodiments are directedto a combination of tailored sectioning and field-based smoothing tofabricate an infill structure for an article that is both sectiontailored and field-smoothed (see, for example FIG. 2C). A field-tailoredlattice can refer to a section tailored lattice, a field-smoothedlattice, a section tailored and field-smoothed lattice, or a latticegenerated by another methodology that integrates a field into the infillgeneration process. For example, a field-tailored lattice may refer to alattice that is fabricated via additive manufacture by tailoredsectioning. a lattice that is fabricated via additive manufacture byfield-based smoothing, a lattice that is fabricated via additivemanufacture by essentially any lattice generation technique thatintegrates a field into the infill generation process to generate anon-uniform, graded infill structure, or any combination of such systemsand methods.

Tailored sectioning, field-based smoothing, and combinations thereofrefer to modified circle packing algorithms that ensures connectivitybetween two non-uniform lattice structures at their interface. That is,these systems and methods can generate non-uniform lattice structureswith guaranteed connectivity that address the connectivity issues thatcan arise when polygon sizes are adapted based on field intensities. Forexample, FIG. 3 illustrates how simply changing the relative size ofhexagons in an infill pattern results in a connectivity issue. Systemsand methods of the present disclosure can generate transition latticepatch linkages that provide a continuous transition interface betweenmultiple different non-uniform lattice structures, which addresses thisconnectivity issue.

In tailored sectioning, the sizes of the packing shapes are variedaccording to a tailored sectioning parameter. The tailored sectioningparameter is indicative of a mapping between a discrete number ofpacking shape sizes (e.g., packing circle radii) and field data (e.g.,intensity value of a physical field over a region). Thresholding candivide the region into sections that have patches of regular polygonstailored to different uniform sizes. The number of discrete polygonsizes can vary from application to application by adjusting thethresholding. Thresholds can be user-selectable, pre-defined in memory,or automatically configured by a processor, e.g., based on the nature ofthe field data, the standard deviation or another statisticalcharacteristic of the field data, or essentially any othercharacteristic of the field data.

That is, by tailoring the sizing of the packing shapes based on thefield intensity into several discrete sizes, the resultant grid willautomatically form multiple uniform sections of regular polygons withtransitions between the sections of different sized polygons includingirregular polygons that ensure a continuous interface between sections.An example of a section tailored infill structure is shown in FIG. 5 ,and discussed in more detail below.

The tailored sectioning method generally produces an embedded complexwith discrete changes in edge lengths based on the underlying field.Field-based smoothing provides a heuristic for obtaining a morecontinuous change in cell size that is influenced by the field data. Inessence, field-based smoothing attempts to embed each vertex of thecomplex as a circle with a radius that is determined from the field datain such a way that neighboring circles are tangent. For stress fields,regions of higher stress produce circles of smaller radius in order tocreate a denser grid, while regions of lower stress produce circles oflarger radius. The mapping of stress (or other field data) to radii canbe defined by the user. In some applications the maximum stress can bemapped to a user set minimum radius, the minimum stress to a user setmaximum radius, and values in-between can be determined by a processorexecuting a linear interpolation or other methodology for intermediatevalues.

Embodiments of the present disclosure can also involve placing boundarycircles onto the boundary of a user-defined (or CAD/slicer defined)polygon, e.g., such that each circle center corresponds to a boundaryvertex that lies on the boundary polygon. That is, systems and methodsof the present disclosure can constrain the circle packing andultimately the infill structure generated to a particular boundary, suchas the infill or surface boundary of the part being additivelymanufactured.

Some embodiments of the field-based smoothing heuristic provide thefollowing constraints: (1) that neighboring circles be tangent, (2) thatboundary circles lie on the boundary of a polygon, and (3) that radiiconform to the specified field. In this way, field-based smoothing issignificantly over-constrained. In alternative embodiments, just (1) and(2) or (1) and (3) together are sufficient to provide a variantheuristic. In essence, the field-based smoothing heuristic involvestreating the network similar to a spring network and iterativelysearching for an equilibrium to allow partial satisfaction all of theconstraints. The equilibrium criteria can vary from application toapplication. In some embodiments, it may be desirable to ensure aboutequal satisfaction among all criteria in order to spread errors fairlyevenly across the network, which ultimately can result in an improvementin static load bearing or other metrics of a 3D printed article. Inother embodiments, certain constraints can be weighted higher or lowerthan others such that the equilibrium point is skewed toward aparticular constraint(s).

Details regarding various exemplary embodiments of these aspects of thepresent disclosure are discussed in detail below in connection withsystematic generation of lattice structures and applications to smalland large-scale additive manufacturing.

Some embodiments of the present disclosure are generally directed to asystem and method to generate a non-uniform graded polygonal structure,or representation in memory thereof, based on a selected field thatdefines one or more regions having a particular effect (e.g., multipleregions with different levels of internal stress or different thermallevels). For example, some embodiments provide a non-uniform, gradedhoneycomb structure based on a given field (e.g., an internal stressprofile or a thermal profile).

Some embodiments of the present disclosure are generally directed to anon-uniform graded polygonal structure that is locally scaled accordingto a field to accommodate different size meshes. For example, oneembodiment can generate an infill structure with a coarse meshcorresponding to a low stress area and a fine mesh corresponding to ahigh stress area. However, attaching a coarse mesh to a fine meshpresents a connectivity issue at the interface between the two differentlattices because they are not guaranteed to align. This can be addressedby locally scaling the size of the mesh at the interface. For example,the structure can include multiple different size meshes and the variousembodiments of the system and method ensure satisfactory connectivitybetween the different sized meshes. In one embodiment, the non-uniform,graded polygonal structure is scaled to two different size meshes, acoarse mesh and a fine mesh.

The systems and methods of the present disclosure can generate additivemanufacturing instructions, e.g., G-Code, which can be provided to a 3Dprinter to additively manufacture a part or article that includes anon-uniform lattice structure. As an example, an airplane wing with anon-uniform lattice infill structure can be manufactured according tovarious embodiments of the present disclosure.

Infill Lattice Generation

A configuration of tangent circles yields a contact graph T byconnecting the centers of tangent circles with straight line segments.The embodiments of the present disclosure systematically and robustlygenerate circle packings whose contact graphs T form triangulationsfitted to prescribed two-dimensional regions. The graph T can beconverted to its dual graph, denoted G, by connecting centers ofadjacent triangles of T with straight line segments. G is trivalent,meaning that each vertex belongs to three edges. An example of this isillustrate in FIG. 4 , which shows how a packing by uniform sizedcircles yields a regular triangulation T that then generates a dualgraph G, in this case, a regular hexagonal grid. When the circles of thepacking are not of uniform size, the nature of the resulting grid Gchanges. For example, perhaps as best shown in FIG. 5 , a non-uniformcircle packing P yields a non-regular triangulation T that converts to agrid G which is honeycomb-like, with mostly hexagonal cells, but withcell side counts in the range 5-7 (and occasional 4's or 8's).Accordingly, by adjusting the circle packing, the resulting lattice Gremains connected and trivalent, but the sizes of its cells vary basedon the sizes of the circles.

Some embodiments of the present disclosure, such as tailored sectioningembodiments (see FIG. 2A), field-based smoothing embodiments (see FIG.2B), and combined tailored sectioning and field-based smoothingembodiments (see FIG. 2C), involve modifications of circle packingtechniques and their application to additive manufacturing toolpaths.That is, a grid G can be constructed for an infill structure that has ahoneycomb appearance because they are all obtained as the duals tocircle packings. The circles form mutually tangent trips, meaning thatconnecting the centers of tangent circles yields a triangulation T, asappears in FIG. 5 . The hexagonal grid G is realized as the dual graphto T. Embodiments of the present disclosure are generally directedtoward the construction and manipulation of the circle packings P, toextract a grid G.

One familiar (circle) packing is the hexagonal or “penny” packing,involving circles of uniform radius, each surrounded by six tangentneighbors. Such a circle packing is mentioned above and illustrated inFIG. 4 . Packings of the present disclosure display more variety butretain the feature that the circles come in mutually tangent triples. Inembodiments of the present disclosure, P denotes a packing, meaning acollection P={C_(v)} of circles in a plane with the property that whenwe connect the centers of tangent circles in P we obtain a planartriangulation graph. The graph is termed the “triangulation” for P anddenoted T=T(P). For each vertex v of T, there is a corresponding in P.Write v˜w if v and w share an edge in T, meaning C_(v)˜C_(w), i.e., thatC_(u) and C_(w) are tangent. The dual graph, denoted G=G(P), is our realtarget. In the case of uniform hexagonal packings, for instance, G is ahoneycomb pattern of six-sided cells.

Combinatorics will generally not be hexagonal. If V, E, and F denote thenumber of vertices, edges, and faces of the triangulation T, the Eulercharacteristix χ(T)=V−E+F will always be one, meaning T triangulates atopological disc. “Boundary” vertices and edges are those on theperiphery of T, denoted T_(∂), while the other vertices and edges aretermed “interior,” denoted T_(int). Denote by N(v) the set of neighborvertices of v, N(v)={u: u˜v}. The degree of v, deg(v), is thecardinality of N(v). In the hexagonal case, deg(v)=6 for all v∈T_(int),but in general, degrees will fall in the range 5-7, with thepreponderance being 6 and with occasional 4's and 8's.

To compute a packing P={C_(v)}, a processor computes R={r_(v)}, theassociated radii of the circles, and Z={z_(v)}, the associated circlecenters. It may be counterintuitive, but the process starts withtriangulation T. The processor then computes the radii r_(v), typicallytaking the radii of vertices in the boundary as initial data. Finally,with the combinatorics and the radii in memory, the processor cansuccessively compute circle centers z_(v). Standalone packing enginesare readily available and can handle simple and complex packings alike.

Although some embodiments of the present disclosure leverage the circlepacking paradigm, the constructions involve compromises. Typicalboundary conditions involve boundary radii, centers, and/or boundaryangle sums. In certain embodiments, a border may or may not be includedaround a shape, and if included, it may be defined irrespective of theinterior grid. Also, as sizes are modified as part of tailoredsectioning (see below), tangency compromises can be made by bringing in“inversive distance” parameters.

In general, each cell of the grid G roughly circumscribes the associatedcircle of the packing P. Accordingly, these cells are “almost round”rather than distorted. The number of edges of the cell associated with vis deg(v).

As T is a triangulation, the circles of the packing P form triples. Thismeans that the corners of the cells of the grid G are triple points,points incident to three edges. This and the fact that these edges meetwith roughly equal angles, can provide clear structural advantages.

There is a mathematical rigidity attached to the packing P. For example,in the infinite hexagonal case, every circle of the packing P must havethe same radius—if one radius of its packing P is changed, it isimpossible to compensate with other radii adjustments to maintain ahexagonal circle packing in a way that is not simply ascaled/translated/rotated version of the original packing. Although theembodiments of the present disclosure deal with practical finite andnon-hexagonal triangulations, this notion of rigidity persists: Onemanipulates packings for triangulation T by manipulating boundaryconditions, but once those boundary conditions are set, the geometry ofthe packing P (and hence, the geometry of the grid G) is uniquelydetermined.

Each cell of the grid G is associated with a single number, the radiusof its circle. This makes for easy computations and avoids degeneraciesand accounts for the “conformal” nature exhibited by circle packings.

Tailored Sectioning Infill Lattice Generation

A region Ω in an x, y-plane is presented, for which an infill grid G isdesired. The grid G can be obtained by generating a circle packing P inregion Ω and extracting the grid G as its dual grid. From the circlepacking P an underlying triangulation T can be obtained, for example,for use in subsequent field-based smoothing. Optionally, elements may beincluded in the circle packing P, triangulation T, and grid G associatedwith the boundary of the region Ω.

One goal of tailored sectioning is to accommodate additional constraintson the grid G represented by a scalar field to which the grid Gresponds. This field, specified by a non-negative function ƒ(x, y) onthe region Ω, may represent a distribution of stress, weight, or someother physical property that varies across the region Ω. Put anotherway, one goal of tailored sectioning is to align the granularity of thegrid G with the values of ƒ: where ƒ is larger, the cells of G should besmaller and vice versa. This can be accomplished by configuring aprocessor to adjust the granularity of the circle packing P.

In a simple case where ƒ is essentially constant, the circle packing Pcan be created by cookie-cutting the shape of the region Ω out ofregular hexagonal circle packing of the plane. Such a case isessentially illustrated in FIG. 4 . The processor can be configured toaccept a selection from a user interface as to a common radius of thecircles and relative position of the circles within the region Ω tooptimize the circle packing P—for instance, so that a row of its circleslies along a given edge of the region Ω. Circles lying along a givenedge of the region Ω can include lying tangent the edge of the region,lying such that the center of the circles lie on the edge of the region,or essentially any other configuration where the circles either overlapthe edge of the region or are in close proximity to the edge of theregion. In some embodiments the circles can be packed to lie along theedge of the region in substantially the same manner, in otherembodiments the circles can be packed such that they lie along the edgeof the region in different manners.

In other cases, the values of ƒ will vary across the region Ω. Thecircle packing P can be generated out of circles that vary in radius.For example, the circles of the circle packing P can be smaller wherethe magnitude of the field ƒ is larger and vice versa. Examples of thisare represented in FIG. 1 and FIGS. 9A-D. This can be accomplished whilemaintaining or substantially maintaining local uniformity to the extentpossible, by utilizing a limited number of distinct radii for thecircles so that the circles of the packing P form local hexagonalpatches. And, due to using a limited number of distinct radii, the gridG will have generally uniform cells within the local patches, butirregular cells in the transition zones between patches.

This system and method of tailored sectioning provides flexibility thatis not present in conventional lattice generation. The method can bedescribed as a series of steps including: obtaining a field function ƒthat maps a region Ω into an interval [a, b]⊂

⁺, selecting a number m of decreasing values b=ƒ₀>ƒ₁>ƒ₂> . . .>_(fm-1)>ƒ_(m)=a, selecting increasing radii 0<r₁<r₂< . . .<r_(m-1)<r_(m) where r₁ and r_(m) represent the radii of the largest andsmallest circles, respectively, permitted in the circle packing P, andselecting a micro-lattice parameter s and a micro-lattice M=M(s). Heres>0 is such that each circle radius r_(j) is roughly equal to an integermultiple n_(j) of s. For example, given ϵ>0, the processor can beconfigured to select s suitably small that there will exist integers0<n₁<n₂< . . . <n_(m) with |n_(j)s−r_(j)|<ϵr_(j), j=1, . . . , m. Theassociated micro-lattice M=M(s) is a regular hexagonal lattice withlattice spacing 2s (the distance between neighboring lattice points),with a convenient orientation and juxtaposed with the region Ω. For eachinteger n>1 there are superlattices M_(n) within M that are regularhexagonal lattices with lattice spacing n(2s). For each j the processorcan be configured to select and fix such a superlattice M_(j)=M_(n) _(j). This family {M_(j)} of chosen superlattices are held in memory as thebasis for construction of the circle packing P. In some embodiments, forj<m, the processor can be configured to include additional circles ofradius r_(j) to help smooth the transition to circles of the next largerradius r_(j+1).

The nearest neighbor triangulation T of the centers of the circlepacking P can be defined. In addition, the grid G can be defined as theconcrete dual grid to the circle packing P. Referring to FIG. 6 anexample of a grid G, with blowups showing an exemplary triangulation andcircle packing P are depicted. The cells of the grid G are locallyuniform honeycombs in regions between the field function ƒ_(j)-levelsets of the field F. There are irregular cells between those regions,though typically having no less than four and no more than eight sides.Tangency between circles can be generalized to accommodate overlaps orseparations between neighboring circles. This can be done with“inversive distance” labels on the edges of T. Further optimization canbe obtained with field-based smoothing, which will addressirregularities in the cells of the grid, as well as irregularities incells between the grid and perimeter edges around ∂Ω.

A slicer software program can conduct this process sufficiently fastthat a user can cycle through many repetitions with various parameters,such as the micro-lattice parameter s, the field function values ƒ_(j)and the integers n_(j), to optimize the grid G. to adjust the totalweight of the infill material, to further tailor the gradations of cellsize, or to incorporate ad hoc adjustments in local areas.

Field-Based Smoothing Lattice Generation

The tailored sectioning system and method described above can produce anembedded complex with discrete changes in edge lengths based on anunderlying field, such as a stress, thermal, or essentially any otherspecified field. The field-based smoothing is a heuristic for obtaininga more continuous change in cell size that is influenced by the fielddata. An evolution of a complex starting from an initial planarembedding with radii set to 1 is illustrated in FIG. 7 , with the fourrepresentations illustrating the state of the packing at iterations 0,100, 200, and 300.

Field-based smoothing attempts to embed each vertex of the complex as acircle with a radius that is determined from the field in such a waythat neighboring circles are tangent. Regions of higher field values(e.g., more stress) produce circles of smaller radius in order to createa denser grid, while regions of lower field values (e.g., less stress)produce circles of larger radius. Although the convention chosendescribes higher field values representing more stress and lower fieldvalues representing less stress, the mapping of stress to radii can bedefined by the user. For example, in some embodiments, maximum stress ismapped to a user set minimum radius, and the minimum stress is mapped toa user set maximum radius, and linear interpolation between the twovalues provides intermediate stress values. Field-based smoothing canalso attempt to place the boundary circles of the disk onto the boundaryof a user-defined polygon, meaning that each circle center cancorresponding to a boundary vertex that lies on the user-definedpolygon.

Summarizing, some embodiments of the field-based smoothing heuristicprovide the following constraints: (1) that neighboring circles betangent, (2) that boundary circles lie on the boundary of a polygon, and(3) that radii conform to the specified field. In this way, field-basedsmoothing is significantly over-constrained. In alternative embodiments,just (1) and (2) or (1) and (3) together are sufficient to provide avariant heuristic. In some aspects, the field-based smoothing heuristicinvolves treating the network similar to a spring network anditeratively searching for an equilibrium to allow the field-basedsmoothing to partially satisfy all of the constraints thereby spreadingerrors fairly evenly across the network, which ultimately can result inan improvement in static load bearing or other metrics of a 3D printedarticle.

The rigidity of the circle packing allows implementation of a refinementon the packing that maintains the original constraint of the face anglesum of the boundary vertices in the triangulation. An angle sum is thetotal angle α(v) around a vertex v, and the constraints on the packingdefined by the radii of the boundary vertices can also be formulated bythe face angle sums of the boundary vertices. The radii of a subset ofthe circles in the packing can be changed without altering theconstraints of the face angle sums along the boundary.

This rigidity can also lead to unexpected changes in the combinatoricsof the packing by slight changes in the boundary conditions imposed.Refinement can be achieved based on simulated field values imposed onthe target object or article.

Field-based smoothing can be described in three stages: (1) adescription of the heuristic for adjusting a given planar grid embeddedwith an initial circle set to a nearby configuration in which allneighboring circles are tangent; (2) a description of how to add theconstraint of placing vertices on the boundary of the polygon andoptionally pinning some vertices to polygon corners; and (3) adescription of how to incorporate field values to adjust radii to moredensely pack regions of higher values (e.g., higher stress) while stillattempting to maintain neighbor tangencies.

Satisfying Neighbor Tangencies

Embodiments of the systems and methods of the field-based smoothingheuristic can generally accept as inputs a complex K (e.g., such as atriangulation T obtained from tailored sectioning) and a packing Ptogether with an initial placement of vertices as circles in a plane.Each vertex v corresponds to a circle C_(v) centered at p(v) and radiusR(v). N(v) denotes a set of neighbors of vertex v in complex K, and edgevector E(v, v′)=p(v′)−p(v) denotes the edge vector from v to v′, anddistance d(v, v′)=∥E(v, v′)∥ denotes the distance between circle centerscorresponding to v and v′.

Field-based smoothing works, in general, by first updating the positionp(v) and then updating the radius R(v) for each vertex v independentlyof the other vertices. Consider a vertex v and one of its neighbors v′.The position of a circle C(v) can be corrected by moving it towardsC(v′) along an imaginary line connecting the two centers p(v) and p(v′)until the two circles become tangent. This can be referred to asposition correction of C_(v) towards C_(v′). The position correction isgiven by

$\begin{matrix}{{P\left( {v,{v'}} \right)} = {\frac{{d\left( {v,v^{\prime}} \right)} - {R(v)} - {R\left( v^{\prime} \right)}}{d\left( {v,v^{\prime}} \right)}{{E\left( {v,v^{\prime}} \right)}.}}} & \end{matrix}$The process also includes computing a radius correction of C_(v) towardsC_(v′) which is the change in radius to make C_(v) tangent to C_(v′)without changing its position. The radius correction is given by: p(v,v′)=d(v, v′)−R(v′). Then, in this embodiment of the field-basedsmoothing heuristic, the processor is configured to compute the averageposition and radius correction values over all neighbors:

$\begin{matrix}{{{{P(v)} = {\sum\limits_{v^{\prime} \in {N(v)}}{P\left( {v,v^{\prime}} \right)}}},{and}}{{\rho(v)} = {{\frac{1}{❘{N(v)}❘}{\sum\limits_{v^{\prime} \in {N(v)}}{\rho\left( {v,v^{\prime}} \right)}}} - {{R(v)}.}}}} & \end{matrix}$

Finally, in this embodiment, the update in a single iteration to avertex v's position p(v) and radius R(v) is given by:p(v)_(new) :=p(v)+δP(v), andR(v)_(new) :=R(v)+δρ(v)

The parameter δ is a user-defined value that controls how big the updatestep is at each iteration. Larger values may become unstable whilesmaller values will take a larger number of iterations to converge. Someembodiments use δ=0.01.

The iterative heuristic can apply the updates above to each vertex inthe complex K. In this embodiment, a total number of iterations can bespecified as a user-controlled parameter. FIG. 7 shows an example offield-based smoothing starting with an initial grid with a small initialcircle placed at each vertex. The four grids represent an evolution of acomplex K starting from an initial planar embedding with radii set to 1.The four images represent the state of the packing at iterations 0, 100,200, and 300.

Placing Boundary Circles on the Boundary of a Polygon

In the current embodiment of the field-based smoothing, a user can pushthe boundary circles outwards to the boundary of a user defined polygonwith a user interface, such as a mouse or touch screen in communicationwith the computer performing the field-based smoothing. Alternatively, aprocessor can be configured to execute a program stored in memory thatautomatically pushes the boundary circles outward to the boundary of auser-defined polygon according to a set of criteria. Consider a boundaryvertex v and let

denote the point on the user defined polygon nearest the circle centerp(v).

can be incorporated as an additional attraction point for the positioncorrection calculation:

$\begin{matrix}{{\left. {{P(v)} = {- {p(v)}}} \right) + {\sum\limits_{v^{\prime} \in {N(v)}}{P\left( {v,v^{\prime}} \right)}}},} & \end{matrix}$

This essentially has the effect of moving the boundary vertices onto theboundary of the polygon. In some embodiments, a user can optionallyselect a corner of the polygon with a user interface instead of usingthe nearest polygon point as

once for each polygon corner. This has the effect of having the boundaryof the final grid match the boundary of the polygon more precisely,perhaps even exactly. FIGS. 8A-B show an example of placing boundarycircles on the boundary of an input polygon (FIG. 8A) and with certainvertices pinned to the corners of the polygon (FIG. 8B). Specifically,in FIG. 8A, the boundary circle centers are attracted to the boundary.In FIG. 8B, the boundary circle centers are attracted to the boundaryand midway through the computation, the seven circles closest to theseven corners of the polygon are “pinned” to the polygon vertices.

Incorporating the Field Data

Two embodiments of methods for incorporating scalar field data (e.g.,scalar stress field data) will now be described and compared inconnection with FIG. 9 .

For both embodiments, F(p) denotes the value of the scalar field at aparticular point p, while the minimum and maximum values are denoted by

${F^{-} = {\min\limits_{p}{F(p)}}}{and}{F^{+} = {\max\limits_{p}{{F(p)}.}}}$R⁻ and R⁺ denote user specified minimum and maximum desired radii. Theprocessor is configured to associate the minimum field value of F⁻ withthe maximum desired radius R⁺, the maximum field value of F⁺ with theminimum desired radius R⁻, and linearly interpolate between the two forpoints whose field value F(p) is in-between. Thus, the desired radiusfunction:

${{D(p)} = {\frac{\left( {F^{+} - {F(p)}} \right)\left( {R^{+} - R^{-}} \right)}{F^{+} - F^{-}} + R^{-}}}.$

First Embodiment: To incorporate the field data into the iterativemethod described above, the radius correction function is altered to

${\rho(v)} = {\frac{1}{1 + {❘{N(v)}❘}}{\left( {\left( {{\sum\limits_{v^{\prime} \in {N(v)}}^{}{\rho\left( {v,v^{\prime}} \right)}} - {R(v)}} \right) + \left( {{D\left( {p(v)} \right)} - {R(v)}} \right)} \right).}}$

This incorporates the desired radius as another member of averagechange. In this embodiment, the desired radius has a small influence onthe overall sum. Thus, for a user to weight the desired radius morehighly against the neighbor tangency computation, the user has toexaggerate the desired radius computation. This can be done bymultiplying R⁺ by a multiplicative factor larger than 1 or bymultiplying R⁻ by a multiplicative factor between 0 and 1 to force thesmaller circles to get even smaller.

Second Embodiment: Another method of achieving a similar effect, whichcan be incorporated into the lattice generation software is to modifythe update to

$\begin{matrix}{{\rho(v)} = {\in {\left( {{D\left( {p(v)} \right)} - {R(v)}} \right) + {\left( {{1 -} \in} \right)\frac{1}{❘{N(v)}❘}{\sum\limits_{v^{\prime} \in {\backslash{N(v)}}}^{}{\left( {{\rho\left( {v,v^{\prime}} \right)} - {R(v)}} \right).}}}}}} & \end{matrix}$The user-selected parameter ϵ controls how strongly the heuristic shouldfavor the desired radius over tangency. Since vertices have nearlyconstant valency these embodiments have a similar effect; however, thefirst embodiment asks the user to exaggerate the desired radius in a waythat may be counter intuitive, while the second embodiment allows theuser to simply control how much the desired radius influences theoverall computation using a single parameter ϵ which may be more userfriendly in practice. A comparison of the two embodiments is illustratedin FIGS. 9A-D on a grid that is influenced by a stress field, butwithout the boundary polygon constraints from the last section.

Referring to FIGS. 9A-D, a comparison of the two embodiments forincorporating the stress field is shown. FIG. 9A illustrates the stressfield being incorporated with darker shading representing higher stressvalues (i.e., larger intensity values of field data) and lighter shadingrepresenting lower stress values (i.e., lower intensity values of fielddata). The field data can essentially be taken from any field and storedin memory. It can be obtained by or provided to the processor executingthe slicing software program that is configured to generate additivemanufacturing instructions for the infill structure. For example, thefield data can be simulated, actual, or user-specified. For example, astress field (or other type of field) can be simulated by an internalstress simulation program that analyzes CAD or other representation ofthe article being fabricated. As another example, a thermal field (orother type of field) can be measured with a suitable sensor. Themeasurements and simulation may be generated based on a differentversion of the article. For example, a different additively manufacturedversion, perhaps with uniform honeycomb or some other infill structure.Or, from a version of the article manufactured by a differentnon-additive manufacturing method. As yet another alternative, the fielddata may be user specified and not based on simulated or actual fielddata. It should also be understood that the field data may not includeintensity values for the entire area of the field corresponding to thearticle. For example, the field data might include some minimum and/ormaximum values corresponding to locations on the part with otherintensity values being interpolated. The field data may represent aphysical field that the resultant additively manufactured part isexpected to be subject to, such as internal stress under certainexpected loading conditions, certain thermals under expected temperatureexposure levels, or essentially any other field the part is expected toexperience. The field data may or may not be available at the layerlevel of the article. In instances where the field data is available bylayer, the layer field data can be utilized. However, in instances wherelayer field data is not available, surface field data may be applied toeach layer. Such data may be applied as is, assuming substantiallysimilar field experience for each layer, or an interpolation method maybe utilized based on surface values and relative position of the subjectlayer.

Referring back to the specific field-smoothed embodiments depicted inFIGS. 9B-D. Both embodiments start from the same initial state (i.e.,complex K) as shown in FIG. 7 . An example of the first embodiment witha minimum circle radius of 8 and maximum circle radius of 60 isillustrated in FIG. 9B. FIG. 9C illustrates an example of the secondembodiment with a minimum circle radius of 10, maximum circle radius of50, and user parameter ϵ=0.75. FIG. 9D illustrates another version ofthe second embodiment, this one with a minimum radius of 10, maximumradius of 50, and user parameter of ϵ=0.95. The units for the circleradii can be any suitable unit depending on the application and scale ofthe desired infill structure. For example, the units may refer tocentimeters, millimeters, or another measurement unit appropriate forthe application and scale of the article being fabricated.

FIGS. 10A-C show examples of incorporating both the stress field and thepolygon boundary condition constraints for both embodiments discussedabove. Specifically, the FIG. 10A illustrates a grid generated by thefirst embodiment with a desired minimum radius of 8 and maximum radiusof 60. In this case, the underlying stress field contributes almost nochange from the unstressed case of FIG. 7 . FIG. 10B illustrates anotherversion of the first embodiment but with the minimum radius set to −50.This exaggeration allows the system to better incorporate the stressfield. FIG. 10C illustrates an exemplary complex with minimum radius 10,maximum radius 50 and ϵ=0.95. This achieves a qualitatively similareffect as the FIG. 10B complex, but without the using a negative radiusinput as an exaggeration.

Infill Lattice Generation Topology

Two graphs G₁ and G₂ are isomorphic if there is a bijection ƒ:V(G₁)→V(G₂) mapping the vertex set of G₁ to the vertex set of G₂ suchthat uv is an edge of G₁ if and only if ƒ(u)ƒ(v) is an edge of G₂.Colloquially, two graphs are isomorphic if they are re-labelings orre-drawings of each other. The isomorphism class of a graph G is the setof all graphs that are isomorphic to G.

With respect to tailored sectioning and field-based smoothing, they cangenerate graphs whose dual graphs are isomorphic to a subgraph of thehoneycomb (hexagonal) lattice. For example, FIG. 11 illustrates anun-deformed graph, Graph A, that can be deformed utilizing one of theembodiments of the present disclosure into a deformed graph, Graph B,illustrated in FIG. 12 . The dual graphs, DualGraph A and DualGraph B,of Graphs A and B are shown in FIGS. 11-12 . DualGraph A is isomorphicto DualGraph B and likewise, DualGraph B is isomorphic to DualGraph A.It is worth noting that Graph A is not isomorphic to DualGraph A orDualGraph B.

One difference between tailored sectioning and field-based smoothing isthat field-based smoothing allows for more precise control of theisomorphism class of the graph that is generated. The field-basedsmoothing maintains the same isomorphism class for the underlying graphthroughout the process (meaning that no vertices, edges, or faces in thegraph are added or removed). This means that if the dual graph of theinitial grid is hexagonal and isomorphic to a subgraph of the hexagonlattice, then the dual graph of the final grid will also be hexagonaland isomorphic to a sub-graph of the hexagonal lattice.

Small-Scale Additive Manufacturing Example

The systems and method for lattice generation can be applied to generatean infill structure for a 3D printed article, such as an airplane wing.FIGS. 4A-C illustrate three exemplary wings that can be manufactured inaccordance with embodiments of the present disclosure.

Field Data

The field data can be representative of essentially any physicalcharacteristic of the part being additively manufactured. Internalstress field and thermal fields are two practical applications, butessentially any field that varies over the extent of the part can havepractical application in connection with the embodiments of the presentdisclosure. In many practical applications, field data will includevalues representative of physical characteristics, such as thermal orstress characteristics. The field data can be obtained from essentiallyanywhere. For example, field data can be communicated over a networkfrom a database or server having a repository of such data, determinedexperimentally by use of sensors on a prototype, duplicate, or otherphysical representation of the target part, or via simulation based oncharacteristics of the target part.

In one embodiment, a static loading or other type of simulation can beperformed, e.g., via finite element analysis (FEA). Such a simulationcan be performed with commercial FEA software, such as Abaqus 2018 orother FEA software. Although such a simulation can produce a stressfield that can be utilized in connection with the embodiment of thepresent disclosure, it should be understood that the lattice generationsystems and method of the present disclosure can accept a field from anytype of loading case or a combination of multiple loading cases,provided that the output of the simulation is presented in atwo-dimensional field.

Infill Lattice Generation

Embodiments of tailored sectioning and field-based smoothingmethodologies can be adapted for use in additive manufacturing slicersoftware to generate an infill lattice structure for a part to beadditively manufactured. For example, referring to FIG. 2A, anon-uniform honeycomb infill structure with two distinct size hexagonpatterns can be generated with slicer software incorporating anembodiment of the tailored sectioning. Referring to FIG. 2B, anexemplary non-uniform honeycomb infill structure with gradually gradedhexagons can be generated with an embodiment of the field-basedsmoothing. As another example, referring to FIG. 4C, a non-uniformhoneycomb infill can be generated based on a combination of embodimentsof tailored sectioning and field-based smoothing methodologies. Theaverage hexagon size was calibrated so that the infill of each wing hasan equal weight of about 74 g.

The infill structures of FIGS. 4A-C can be manufactured in a small-scale3D printer, such as a Stratasys Fortus 400MC, with ABS plastic filament.Stiffness of infill lattice structures generated with the systems andmethods of the present disclosure can be evaluated with static loadtesting. Generally, load weight and deflection show a linearrelationship. Field-based smoothing tailored sectioning methodologiesgenerally produce infill lattices that deflect less relative to uniforminfill lattice structure counterparts for the same load, with infilllattices generated by a combination of tailored sectioning andfield-based smoothing deflecting even less.

Through calibration of the parameters in both methods (Smoothing andSectioning), the disparity of the pattern size can be increased in orderto improve stiffness. Field-based smoothing may be better for certainapplications and tailored sectioning may be better for certainapplications. Field-based smoothing generates a lattice structure withgradually changing unit size, and the disparity of the unit size can bechanged via calibration and parameter selection (i.e., differencebetween small hexagon and large hexagon). Tailored sectioning cangenerate an infill lattice structure with an abrupt change in the unitsize, which can be desirable in some applications. However, it is alsopossible with tailored sectioning to partition an area into multiplesections and assign slightly larger or smaller unit circle sizes fromone section to the next section, which can provide a non-uniformhoneycomb lattice with more visually gradual changes, and provide agreat degree of control.

Large-Scale Additive Manufacturing Example

Embodiments of the present disclosure can also be utilized inlarge-scale additive manufacturing. The geometry and the dimensions ofan exemplary flat wing for large-scale additive manufacturing printingare shown in Error! Reference source not found. A finite elementanalysis (FEA) can be performed on the flat wing design, with suitableboundary conditions applied to the flat wing. Further constraints can beapplied, such as pressure constraints and elastic material properties. Avon Mises stress field, or other type of stress field, can be obtainedfrom the FEA simulation as shown in FIG. 17B.

By way of example, a uniform lattice structure in the domain of the wingcan be generated as shown in FIG. 14 . An optimized lattice structure180 can be generated with both the tailored sectioning method and thefield-based smoothing method. The width of the lattice ribs in this caseis set to about 0.26 inches (˜6.6 mm), which is two-bead widths of theextruded material from the nozzle diameter of 0.1 inch without a tamper.The boundary line of the wing can be added to the lattice structure toclose the incomplete polygons at the boundary. The width of the boundaryline can be set to the same width of the lattice ribs (6.66 mm) or adifferent value. The addition of the boundary line can be done usingcomputer-aided design (CAD) software. The size of the unit cell (in thiscase, the hexagon cell) of the uniform lattice was calibrated such thatthe total weight of the uniform lattice wing matches the total weight ofthe optimized lattice wing. In the generated optimized lattice, thehexagon unit size decreases as the location moves down from the tip ofthe wing to the root of the wing. The gradual decrease in the cell sizefrom hexagon cell 182 to hexagon cell 184 and from hexagon cell 186 tohexagon cell 188 results from application of the field-based smoothingmethod. The abrupt change in the cell size from hexagon cell 184 tohexagon cell 186 results from application of the tailored sectioningmethod. FIG. 14 shows an exemplary comparison of the cell sizes betweenthe uniform lattice 190 and optimized lattice 180, specifically oneexemplary lattice cell 192 of the uniform lattice 190 is depicted nextto several exemplary lattice cells 182, 184, 186, 188 of the tailorsectioned and field smoothed lattice 180. In this exemplary embodiment,it is worth noting that the two lattices 180, 190 have about the sameweight (1.1 kg), and the hexagon size of the uniform lattice (71.5 mm)is in between the size of hexagon cell 184 (89.3 mm) and the size ofhexagon cell 186 (42.4 mm) in the optimized lattice as shown in FIG. 14.

Put simply, FIG. 14 illustrates a wing design with a uniform lattice anda wing with an optimized lattice fabricated in accordance with thetailored sectioning and field-based smoothing embodiments of the presentdisclosure. FIG. 14 also shows a comparison of the hexagon size from theuniform lattice and the optimized lattice. In this case, the articlematerial is acrylonitrile butadiene styrene (ABS) reinforced with 20%wt. carbon fiber, though essentially any other suitable material couldbe used for the fabrication. A nozzle diameter of 0.1 inch (2.54 mm) andlayer height of 0.05 inch (1.27 mm) was used in this embodiment, thoughdifferent values could be used within the scope of embodiments of thepresent disclosure. The print includes ten layers, though other printsin accordance with the disclosure may include additional or fewerlayers.

Additive Manufacturing System Example

Embodiments of the present disclosure can be utilized in connection withan additive manufacturing system. One exemplary additive manufacturingsystem 100 in accordance with one embodiment of the present disclosureis illustrated in FIG. 15 . The additive manufacturing system 100generally includes a computer 102 and an additive manufacturing machine104. Computers and additive manufacturing machines are generally wellknown and therefore will not be described in detail. Suffice it to say,a computer or processor 102 can essentially be any hardware orcombination of hardware, local, remote, or distributed, capable ofreceiving a representation of an object 116, executing a slicingalgorithm 108 on the representation 116 in order to generate a toolpath110 and additive manufacturing instructions. The algorithms can bestored in memory, individually or collectively, as instructions forexecution by a computer processor. For example, the slicer and toolpathgeneration algorithms may be referred to collectively as a slicersoftware program that outputs (e.g., stored in memory or communicates toanother module or apparatus) additive manufacturing instructions.

Further, the system can include an additive manufacturing machine 104that can be essentially any suitable additive manufacturing equipmentthat can generate an additive structure according to additivemanufacturing instructions generated by the slicer software program. Forexample, for deposition based additive manufacturing systems, inoperation, the computer 102 receives a representation of an object 116and a slicer 108 slices the model 116 and generates a toolpath 110 forsuccessively additively manufacturing each layer (e.g., by deposition ofmaterial from a nozzle that moves about a print area according to theinstructions). The output of the programs can ultimately be provided inthe form of additive manufacturing instructions to the additivemanufacturing machine 104. The slicer 108 and toolpath generator 110 canbe separate parts of one software program or can be stand-alone softwareprograms that execute on the computer and can communicate directly witheach other or indirectly, for example via files stored in memory on thecomputer. The controller 114 of the additive manufacturing machine 104controls the deposition nozzle 112 according to the instructions toadditively manufacture the object layer by layer. Although the additivemanufacturing system 100 described herein is a deposition based systemwith a deposition nozzle, other types of additive manufacturing machinescan be utilized in connection with embodiments of the presentdisclosure. The particular methodology for additively manufacturinginfill structures generated in accordance with embodiments of thepresent disclosure can vary from application to application. U.S. patentapplication Ser. No. 16/750,631, filed on Jan. 23, 2020 to Seokpum, andhereby incorporated by reference in its entirety, describes varioussystems and methods for additive manufacturing with toolpath bridgesthat can be utilized in connection with the embodiments of the presentdisclosure.

Forming an additive structure, such as a lattice infill structure,includes any process in which a three-dimensional build, part, object,or additive structure is formed in successive layers according to one ormore additive manufacturing techniques. The systems and methodsdiscussed herein are suitable for both small and large scale additivemanufacturing. The embodiments are applicable for essentially anyadditive manufacturing systems involving generation of a non-uniforminfill. For example, suitable additive manufacturing techniques for usein conjunction with embodiments of the present disclosure include, bynon-limiting example, direct energy deposition (DED), material extrusion(e.g., fused deposition modeling (FDM)), welding-based systems, materialjetting, binder jetting, powder bed fusion, and essentially any otheradditive manufacturing process.

The additive structure can be formed with essentially any material orcombination of materials used in additive manufacturing. This caninclude additive manufacturing materials now known or hereinafterdeveloped. Suitable materials can include plastics, fiber composites,ceramics, metals, and other materials. For example, thermoplastics,thermosets, rubber, silicone, carbon fiber, and glass fiber, glassfiber-filled ABS, carbon fiber-filled ABS, to name a few differentmaterials suitable for use with the embodiments of the presentdisclosure.

Molecular Dynamics Infill Lattice Generation

Another aspect of the present disclosure is generally directed tosystems and methods for molecular dynamics based infill latticegeneration. In particular, a force balance equation can be used as thefoundation for a system and method for generating non-uniform infilllattice structure based on field data.

The molecular dynamics infill lattice generation of the presentdisclosure is inspired by the Lennard-Jones potential equation

$V = {4{{e\left\lbrack {\left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6}} \right\rbrack}.}}$The equation and graph illustrate an intermolecular pair potential,sometimes referred to as the 12-6 potential. It provides an archetypemodel for realistic intermolecular interactions.

A force balance equation generally has two terms: pushing force (i.e.,repulsive force) and pulling force (i.e., attraction force). A forcebalance equation essentially involves locating nodes (e.g., two nodes adistance r apart) such that the pushing and pulling forces are balancedto meet a particular equilibrium. If nodes are positioned too close, thepushing force become dominant, and the nodes push each other away.However, if the two nodes are positioned too far apart, the pullingforce becomes dominant over the pushing force, and the nodes are pulledtoward each other, or if the nodes are even farther apart, then theoverall force may become very weak and the nodes do not exert anyappreciable force on one another, pushing force or pulling force. Thesethree states are illustrated in the FIG. 16 graph. The vertical line 302illustrates the distance apart the two nodes end up staying. If they aretoo close (i.e., to the left of line 302), the nodes push each other. Ifthey are far apart, they pull each other (i.e., to the right of line302). And, if they are too far apart, the pulling is weak (i.e., on thefar right side of the graph).

With this backdrop, an embodiment of the molecular dynamics based infilllattice generation method will now be described.

The method can begin with a number of nodes being obtained, generated,or defined over a surface representing the infill layer to be generated,e.g., either randomly distributed or regularly distributed. These nodesform a two-dimensional set of input seeds. FIGS. 17-19 illustratedifferent numbers of input seeds (i.e., nodes) that represent atwo-dimensional area of atoms and their output configuration after themolecular dynamics infill generation methodology is applied and thepositions of the nodes are adjusted.

FIG. 17 shows an example of a relatively low number of input seeds for agiven two dimensional area and the resulting positions of the nodesafter the method adjusts the locations of the nodes. That is, after themethod is applied, the output shows the nodes in positions where theforces are balanced. If the nodes are too far apart, they lose theinteraction force. The amount of nodes can vary by application—if toofew nodes are generated as the input seeds, then the nodes tend tocluster in the output, as shown in FIG. 17 .

FIG. 18 shows a larger number of nodes that are representative of atoms.Again, the nodes are randomly or regularly distributed across atwo-dimensional area. When the molecular dynamics based infill latticegeneration method is applied with a sufficient number of nodes, such asthe case with FIG. 18 , the nodes evenly space out from each other inthe output, as shown.

If the number of nodes representing atoms is allowed to fill the entiredomain as shown in FIG. 19 , then the nodes evenly space out with ashorter distance from each other in the output. Because there are alarge number of nodes, when a stress field is imposed, the effect of theinput stress will be less significant.

A modified version of the force balance equation can be utilized thatintegrates field data into the force balancing. For example, an inputstress factor α can control the distance between nodes at a certainarea. This can be accomplished with the following modified Lennard JonesPotential, with alpha representing an input stress factor:

$V = {4{e\left\lbrack {\left( \frac{\sigma}{r} \right)^{p} - {\alpha\left( \frac{\sigma}{r} \right)}^{q}} \right\rbrack}}$

Inclusion of the input stress factor effectively shifts the equilibriumdistance. For example, as shown in FIG. 16 , the equilibrium distanceline 302 shifts to line 304.

The p and q values can be varied depending on the application. In someapplications, the p and q values are set to 8 and 6, respectively.However, in alternative embodiments, the p and q values can be set toother suitable values such as 4 and 2. In general, with lower values,the effect of alpha (input stress factor) becomes more pronounced in thenode distribution, and with higher values, vice versa.

FIGS. 20A-D illustrates an exemplary embodiment of the moleculardynamics infill lattice generation process. The process begins with arepresentation of a stress field, e.g., as shown in FIG. 20A. FIG. 20Aillustrates a stress field with a distribution of intensity field valuesthat represent a high stress near the upper right corner. The stressfield is converted to a two-dimensional node distribution, such asdepicted in FIG. 20B. The node distribution represents the same stressfield by virtual of node density representing higher intensity values ofthe stress field. Triangulation is then applied to the node distributionto generate a triangular graph over the same two-dimensional area, suchas shown in FIG. 20C. The triangle density represents higher stressvalues in a similar fashion as the node density. From there, the dualgraph can be obtained of the triangular graph, which generates agenerally hexagonal graph (with a few other types of polygons with feweror additional numbers of sides. If applied over a surface of an articlelayer for additive manufacture, the hexagonal graph provides a moleculardynamics infill lattice.

FIG. 21A-C provides another example of a hexagonal graph derived from adifferent stress field (not shown). In this embodiment, the stress fieldhas a high intensity stress values at the center of the graph, but hasotherwise normal stress values. The resultant node distribution fromconverting such a stress field is illustrated in FIG. 21A. Thetriangulation of that node distribution is shown in FIG. 21B. And, thedual graph of that triangular graph is shown in FIG. 21C, which resultsin a non-uniform, continuous, and generally hexagonal graph, with sometransition polygons with different number of shapes towards the edges orother regions where the triangle size changes.

A comparison between a node distribution converted from a zero inputstress field and a node distribution converted from an exemplary stressdistribution input for an exemplary wing structure under loading isshown in FIG. 22A-B. In this exemplary stress distribution input, thestress input values range from about 0 to about 6.4e+5 Mises (i.e.,640,000), with an average stress of about 75%. The comparison shows how,for zero stress input, the resultant node distribution is more or lessuniform, but the node distribution that integrates the wing stress fieldhas nodes densely clustered near the bottom of the wing, where thehighest stress correlates, and has nodes sparsely clustered near the topof the wing, where the lowest stress correlates.

Continuing with the stress integrated node distribution from FIG. 22B,FIGS. 23A-B illustrate a triangulated graph obtained from the nodedistribution, as well as the dual (generally hexagon) graph obtainedfrom the triangulation. The straight lines overlaid on the graphs ofFIG. 22B and FIGS. 23A-B show the boundary of the wing 220.

FIGS. 24 and 25A-B illustrate two exemplary lattice infill structuresgenerated with the molecular dynamics infill generation system andmethod. By controlling the number of nodes in the starting nodedistribution, the relative size of the hexagon cells in the ultimateinfill lattice that is generated can be controlled. In general, thelarger the number of nodes in the starting node distribution, thesmaller the size of the hexagons (and other polygons) in the lattice.FIG. 24 illustrates a representative front view of an exemplary infillstructure for a wing generated with a molecular dynamics infillgeneration method using the same stress field of FIG. 22B. FIGS. 25A-Billustrates representative front and perspective views, respectively, ofan infill structure for a wing generated with the molecular dynamicsmethod using the same stress field, but with a larger number of initialnodes in the node distribution than the FIG. 24 node distribution.

Accordingly, the molecular dynamics infill generation method cangenerally be described by the following steps: defining an initial nodedistribution, adjusting the spacing between the nodes to reach,increase, or maximize a force balance equilibrium between the nodes,wherein the force balance equilibrium accounts for field intensityvalues of a field, such as a stress field, triangulating the adjustednode distribution to generate a triangular graph, and dual graphing thetriangular graph to obtain a dual graph representative of an infillstructure corresponding to the field data. The infill structure can bealigned to the boundary of the part being additively manufactured andconverted to additive manufacturing instructions. Because of themolecular dynamics, node distribution, triangulation, and dual graph,the resultant lattice structure will be non-uniform, but generallyhexagonal with transitions between different size hexagons havingdifferent numbers of sides. The infill structure will automaticallyprovide a continuous lattice structure that can be constrained by theboundary of the part and provide vertices that match the part boundary.

Directional terms, such as “vertical,” “horizontal,” “top,” “bottom,”“upper,” “lower,” “inner,” “inwardly,” “outer” and “outwardly,” are usedto assist in describing the invention based on the orientation of theembodiments shown in the illustrations. The use of directional termsshould not be interpreted to limit the invention to any specificorientation(s).

The above description is that of current embodiments of the invention.Various alterations and changes can be made without departing from thespirit and broader aspects of the invention as defined in the appendedclaims, which are to be interpreted in accordance with the principles ofpatent law including the doctrine of equivalents. This disclosure ispresented for illustrative purposes and should not be interpreted as anexhaustive description of all embodiments of the invention or to limitthe scope of the claims to the specific elements illustrated ordescribed in connection with these embodiments. For example, and withoutlimitation, any individual element(s) of the described invention may bereplaced by alternative elements that provide substantially similarfunctionality or otherwise provide adequate operation. This includes,for example, presently known alternative elements, such as those thatmight be currently known to one skilled in the art, and alternativeelements that may be developed in the future, such as those that oneskilled in the art might, upon development, recognize as an alternative.Further, the disclosed embodiments include a plurality of features thatare described in concert and that might cooperatively provide acollection of benefits. The present invention is not limited to onlythose embodiments that include all of these features or that provide allof the stated benefits, except to the extent otherwise expressly setforth in the issued claims. Any reference to claim elements in thesingular, for example, using the articles “a,” “an,” “the” or “said,” isnot to be construed as limiting the element to the singular.

The invention claimed is:
 1. A method for fabricating an article, thearticle configured to experience, during operation of the article, aphysical field having a non-uniform intensity over the extent of thearticle, the method comprising: generating, by a computer system,representations of layers of the article, each layer comprising aninfill portion, wherein a representation of each layer's infill portioncomprises a corresponding a field-tailored lattice having cells withsides of the same thickness, wherein generating the field-tailoredlattice for a corresponding layer's infill portion comprises: (i)circle-packing the infill portion of the layer, such that adjacentcircles are tangentially in contact, and sizes of the circles correlateto values of the intensity of the physical field at the circles'locations, (ii) computing an intermediate lattice having triangularcells, such that vertices of a triangular cell correspond to centers ofthree adjacent circles of the circle-packed infill portion, and (iii)computing the field-tailored lattice having polygonal cells with 4 to 8walls, such that sides of a polygonal cell correspond to segmentsbetween centers of adjacent triangles of the intermediate lattice: andfor each layer of the article, printing, by an additive manufacturingprinter in communication with the computer system, a respectivestructure embodying the corresponding field-tailored lattice in thelayer's infill portion.
 2. The method for fabricating an article ofclaim 1, wherein the field-tailored lattice is a graded honeycomblattice having hexagonal cells of different side lengths, such thatsmaller side lengths correspond to high values of the physical field andlarger side lengths correspond to low values of the physical field. 3.The method for fabricating an article of claim 1, wherein thefield-tailored lattice comprises: two or more patches, each patch havinghexagonal cells of different side lengths among different patches, suchthat smaller side lengths correspond to high values of the physicalfield and larger side lengths correspond to low values of the physicalfield, and one or more transition zones disposed between the patches,each transition zone having polygonal cells with 4, 5, 7, or 8 sides. 4.The method for fabricating an article of claim 3, wherein each patch hashexagonal cells of the same side length within the same patch.
 5. Themethod for fabricating an article of claim 3, wherein (i) circle-packingthe infill portion of the layer further comprises placing boundarycircles on a perimeter of the infill portion.
 6. The method forfabricating an article of claim 5, wherein, when the perimeter hascorners, and wherein (i) circle-packing the infill portion of the layerfurther comprises pinning particular boundary circles to the corners ofthe perimeter.
 7. The method for fabricating an article of claim 1,wherein circle-packing the infill portion of the layer includesincorporating the physical field using at least one of tailoredsectioning, field-based smoothing, and a combination of tailoredsectioning and field-based smoothing.
 8. The method for fabricating anarticle of claim 1, wherein the additive manufacturing printer incommunication with the computer system is at least one of a big areaadditive manufacturing (BAAM) printer and a small-scale additivemanufacturing printer.
 9. The method of claim 1, wherein the physicalfield is at least one of a stress field and a temperature field.
 10. Themethod of claim 1, wherein printing the structure embodying thefield-tailored lattice includes using a nozzle diameter in a diameter ina nozzle-diameter range of 0.05″ to .0.2″.
 11. The method of claim 1,wherein printing the structure embodying the field-tailored latticeincludes extruding material including at least one of a plastic, a fibercomposite, a ceramic, and a metal.
 12. The method of claim 1, whereinthe article is one of a wing, a propeller blade, a turbine blade, abeam, or a toe of an excavator cup.
 13. A computer method of generatingadditive manufacturing instructions additively manufacturing an objectusing tailored sectioning, the computer method comprising: accessing,with a computer system from memory, object geometry data representativeof geometry of the object; slicing, with a slicer software programexecuted by a computer system, the object geometry data into layer datarepresentative of layers of the object, at least one of the layersincluding an object layer boundary and an object layer infill region;accessing, with a computer system from memory, field intensity valuescorresponding to a non-uniform scalar field distribution over the objectlayer infill region; accessing, with a computer system from memory, atailored sectioning parameter indicative of a mapping between the fieldintensity values and two or more circle sizes; circle packing, with acircle packing program executed by a computer system, the object layerinfill region with packing circles to define a circle-packed infillregion, wherein a first subset of packing circles are sized as one ofthe two or more circle sizes and a second subset of packing circles aresized as a different one of the two or more circle sizes by the circlepacking program according to (i) the field intensity values of thenon-uniform scalar field distribution over the object layer infillregion at positions of the packing circles; and (ii) the tailoredsectioning parameter indicative of the mapping between the fieldintensity values and the two or more circle sizes; triangulating, with atriangulation program executed by a computer system, the packing circlesin the object layer infill region to generate an intermediate graph oftriangular cells such that vertices of a triangular cell correspond tocenters of three adjacent packing circles of the circle-packed infillregion, and dual graphing, with a dual graph program executed by acomputer system, the intermediate graph of triangular cells in theobject layer infill region to generate a non-uniform section tailoredlattice grid in the object layer infill region including two or moreuniform sections of hexagonal lattice patches of different sizescorresponding to the tailored sectioning parameter, and one or moretransition zones between the two or more uniform sections of differentsized hexagonal lattice patches including irregular polygons thatprovide a continuous interface between at least two of the two or moreuniform sections of hexagonal lattice patches of different sizes; andconverting the non-uniform section tailored lattice grid to additivemanufacturing instructions for printing, by an additive manufacturingprinter, a respective physical non-uniform section tailored infilllattice structure embodying the non-uniform section tailored latticegrid for the object layer infill region; and storing the additivemanufacturing instructions in memory.
 14. The computer method ofgenerating additive manufacturing instructions for additivelymanufacturing an object using tailored sectioning of claim 13, whereinthe physical non-uniform section tailored lattice is a modifiedhoneycomb lattice having patches of uniform hexagonal cells of differentside lengths connected by the transition zone of irregular polygonalcells, such that the patches of uniform hexagonal cells with differentside lengths have side lengths that correspond to different thresholdintensity values of the non-uniform scalar field distribution over theobject layer infill region.
 15. The computer method of generatingadditive manufacturing instructions for additively manufacturing anobject using tailored sectioning of claim 14, wherein the one or moretransition zones disposed between the patches have polygonal cells with4, 5, 7, or 8 sides.
 16. The computer method of generating additivemanufacturing instructions for additively manufacturing an object usingtailored sectioning of claim 13, wherein (i) circle-packing the objectlayer infill region further comprises placing boundary circles on aperimeter of the object layer infill region.
 17. The computer method ofgenerating additive manufacturing instructions for additivelymanufacturing an object using tailored sectioning of claim 16, wherein,when the perimeter has corners, and wherein (i) circle-packing theobject layer infill region further comprises pinning particular boundarycircles to the corners of the perimeter.
 18. The computer method ofgenerating additive manufacturing instructions for additivelymanufacturing an object using tailored sectioning of claim 13, whereincircle-packing the object layer infill region includes incorporating thephysical field using field-based smoothing.
 19. The computer method ofgenerating additive manufacturing instructions for additivelymanufacturing an object using tailored sectioning of claim 13, whereinprinting the structure embodying the non-uniform section tailored infilllattice structure includes extruding material including at least one ofa plastic, a fiber composite, a ceramic, and a metal.
 20. A computermethod of generating additive manufacturing instructions additivelymanufacturing an object using a field-based smoothing heuristic, thecomputer method comprising: accessing, with a computer system frommemory, object geometry data representative of geometry of the object;slicing, with a slicer software program executed by a computer system,the object geometry data into layer data representative of layers of theobject, at least one of the layers including an object layer boundaryand an object layer infill region; accessing, with a computer systemfrom memory, field intensity values corresponding to a non-uniformscalar field distribution over the object layer infill region;accessing, with a computer system from memory, a field-based smoothingparameter indicative of a mapping between the field intensity values andtwo or more circle sizes; circle packing, with a circle packing programexecuted by a computer system, the object layer infill region withpacking circles to define a circle-packed infill region, wherein thepacking circles are positioned and sized with respect to the objectlayer infill region by the circle packing program according to aplurality of field-based smoothing heuristic constraints including (i)neighboring packing circles are substantially tangent, and at least oneof (ii) a subset of packing circles are boundary circles that lie alonga perimeter of the object layer infill region; and (iii) size of thepacking circles at positions in the object layer infill regioncorrespond to field intensity values from the non-uniform scalar fielddistribution at the same positions in the object layer infill region,wherein the circle packing program iteratively adjusts positions andsizes of the packing circles to search for an equilibrium that causesthe circle-packed infill region to at least partially satisfy thefield-based smoothing heuristic constraints. triangulating, with atriangulation program executed by a computer system, the circle-packedinfill region to generate an intermediate graph of triangular cells suchthat vertices of a triangular cell correspond to centers of threeadjacent packing circles of the circle-packed infill region, and dualgraphing, with a dual graph program executed by a computer system, theintermediate graph of triangular cells in the object layer infill regionto generate a non-uniform field-smoothed lattice grid of hexagonalcells, such that sides of a hexagonal cell correspond to segmentsbetween centers of adjacent triangles of the intermediate lattice: andconverting the non-uniform field-smoothed lattice grid to additivemanufacturing instructions for printing, by an additive manufacturingprinter, a respective physical non-uniform field-smoothed infill latticestructure embodying the non-uniform field-smoothed lattice grid for theobject layer infill region; and storing the additive manufacturinginstructions in memory.
 21. The computer method of generating additivemanufacturing instructions for additively manufacturing an object usingfield-based smoothing of claim 20, wherein the physical non-uniformfield-smoothed lattice structure is a modified honeycomb latticeincluding at least a subset of neighboring hexagonal cells that smoothlytransition from one side length to a different side length.
 22. Thecomputer method of generating additive manufacturing instructions foradditively manufacturing an object using field-based smoothing of claim20, wherein the field-based parameter is a step size indicative of anamount of change at each iterative adjustment to circle size, circleposition, or a combination thereof.
 23. The computer method ofgenerating additive manufacturing instructions for additivelymanufacturing an object using field-based smoothing of claim 20, whereincircle-packing the object layer infill region further comprises pinningparticular boundary circles to corners of the object layer boundary. 24.The computer method of generating additive manufacturing instructionsfor additively manufacturing an object using field-based smoothing ofclaim 20, wherein circle-packing the object layer infill region includesincorporating the physical field using tailored sectioning.
 25. Thecomputer method of generating additive manufacturing instructions foradditively manufacturing an object using field-based smoothing of claim20, wherein printing the structure embodying the non-uniformfield-smoothed infill lattice structure includes extruding materialincluding at least one of a plastic, a fiber composite, a ceramic, and ametal.
 26. A computer method of generating additive manufacturinginstructions additively manufacturing an object using a field-basedsmoothing heuristic, the computer method comprising: accessing, with acomputer system from memory, object geometry data representative ofgeometry of the object; slicing, with a slicer software program executedby a computer system, the object geometry data into layer datarepresentative of layers of the object, at least one of the layersincluding an object layer planar boundary and an object layer infillplanar grid; accessing, with a computer system from memory, fieldintensity values corresponding to a non-uniform scalar fielddistribution over the object layer infill planar grid; accessing, with acomputer system from memory, a field-based smoothing parameter;iteratively adjusting, based on the field-based smoothing parameter, acircle packing configuration, with a field-based smoothing circlepacking program executed by a computer system, of the object layerinfill planar grid to increase neighboring circle tangency, vertices onthe object layer planar boundary, and correspondence between fieldintensity position and circle position on the object layer infill planargrid; triangulating, with a triangulation program executed by a computersystem, the iteratively adjusted circle packing configuration of theobject layer infill planar grid to generate an intermediate graph oftriangular cells such that vertices of a triangular cell correspond tocenters of neighboring circles, and dual graphing, with a dual graphprogram executed by a computer system, the intermediate graph oftriangular cells in the object layer infill region to generate anon-uniform field-smoothed lattice grid of hexagonal cells, such thatsides of a hexagonal cell correspond to segments between centers ofadjacent triangles of the intermediate lattice: and converting thenon-uniform field-smoothed lattice grid to additive manufacturinginstructions for printing, by an additive manufacturing printer, arespective physical non-uniform field-smoothed infill lattice structureembodying the non-uniform field-smoothed lattice grid for the objectlayer infill region; and storing the additive manufacturing instructionsin memory.
 27. The computer method of generating additive manufacturinginstructions for additively manufacturing an object using a field-basedsmoothing heuristic of claim 26, wherein the physical non-uniformfield-smoothed lattice structure is a modified honeycomb latticeincluding at least a subset of neighboring hexagonal cells that smoothlytransition from one side length to a different side length.
 28. Thecomputer method of generating additive manufacturing instructions foradditively manufacturing an object using a field-based smoothingheuristic of claim 26, wherein the field-based smoothing parameter is astep size indicative of an amount of change at each iterative adjustmentto circle size, circle position, or a combination thereof.
 29. Thecomputer method of generating additive manufacturing instructions foradditively manufacturing an object using a field-based smoothingheuristic of claim 26, wherein circle-packing the object layer infillregion further comprises pinning particular boundary circles to cornersof the object layer boundary.
 30. The computer method of generatingadditive manufacturing instructions for additively manufacturing anobject using a field-based smoothing heuristic of claim 26, whereincircle-packing the object layer infill region includes incorporating thephysical field using tailored sectioning.
 31. The computer method ofgenerating additive manufacturing instructions for additivelymanufacturing an object using a field-based smoothing heuristic of claim26, wherein printing the structure embodying the non-uniformfield-smoothed infill lattice structure includes extruding materialincluding at least one of a plastic, a fiber composite, a ceramic, and ametal.